PageRank

PageRank is an algorithm used by the Google Internet search engine to assign a numerical weighting to each web page. It is a trademark of Google. PageRank results from a "ballot" among all the other pages on the World Wide Web about how important a page is. A hyperlink to a page counts as a vote of support. PageRank is a number between 0 and 10. The most popular web pages have a PageRank of 10. The higher the PageRank the better chances this page has for appearing on the first SERP. Google has not disclosed the exact method for determining the PageRank value but it is based on links to the webpage. A page that is linked to by many pages with high PageRank receives a high rank itself. The Google Toolbar displays the current page's PageRank. See the example:

Interesting fact is that PageRank has been named after one of the two Google founders, Larry Page.

PageRank and SERP are related but different terms. SERP stands for Search Engine Results Page and consists of a list of links to web pages displayed as a result of search query. The order of the listed links is determined by their SERP rank. The SERP rank of a web page depends on a relatively large and continuously adjusted set of factors including PageRank.
SEO (Search Engine Optimization) is aimed at achieving high SERP ranks for a website.